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Some Identities of Tribonacci Polynomials

Yogesh Kumar Gupta*, V. H. Badshah, Mamta Singh, Kiran Sisodiya

School of Studies in Mathematics, Vikram University Ujjain, (M. P.), India

Abstract: The Tribonacci polynomial is famous for possessing wonderful and amazing properties. Tribonacci polynomials tn(x) defined by the recurrence relation tn+3(x)=x²tn+2(x)+xtn+1(x)+tn(x) for n≥0 with to(x) =o, t₁(x)=1, t₂(x)=x². In this paper, we introduce some identities Tribonacci polynomials by standard techniques.

Keywords: fibonacci polynomials, tribonacci polynomials, generating function of tribonacci polynomials

Citation: Yogesh Kumar Gupta, V. H. Badshah, Mamta Singh, Kiran Sisodiya, Some Identities of Tribonacci Polynomials. Turkish Journal of Analysis and Number Theory, 4 (1). 20-22. January 2016. doi:10.12691/tjant-4-1-4

1. Introduction

Mathematics can be considered as the underlying order of the universe, and the Fibonacci numbers is one of the most fascinating discovery made in the mathematical world. Among numerical sequences, the Fibonacci sequence has achieved a kind of celebrity status and has been studied extensively in number theory, applied mathematics, physics, computer science, and biology [2]. The Fibonacci numbers are famous for possessing wonderful and amazing properties. A similar interpretation also exists for Lucas sequence. The Fibonacci numbers have been studied both for their applications and the mathematical beauty of rich and interesting identities that they satisfy.

The Fibonacci sequence {fn} of number Fn is defined by the recurrence relation

for,

[picture] | (1.1)

Binet Formula for Fibonacci number is defined by

[picture] | (1.2)

In 1883, catalan, E [2] was considered related set of polynomials which satisfies the recurrence relations.

The name Fibonacci polynomials is also given to the solution of the relation

[picture] | (1.3)

With, investigated by Jacasthal, E [9]. Byrd, p.F [3] defined the Fibonacci polynomials by the recurrence relation.

[picture] | (1.4)

with initial conditions

swamy, M.N.S [15] and Hoggatt, V.E. [9] almost simultaneously defined the Fibonacci polynomials by

[picture] | (1.5)

with .

Here if we put x=1 in we get which if Fibonacci sequence.

Generating function of Fibonacci Polynomials is defined by

Hyper geometric form of generating function of Fibonacci polynomials is

Tribonacci number Tn [10] defined by

[picture] | (1.6)

The Tribonacci numbers are (0, 1, 1, 2, 4, 7, 13, 24…).

The tribonacci polynomial tn(x) [17] satisfies the following recurrence relation:

and to(x)=o, t₁(x)=1 t₂(x)=x².

In this chapter we present some identities of Tribonacci Polynomials by standard methods.

2. Tribonacci Polynomials

The Tribonacci polynomial tn(x) [17] satisfies the following recurrence relation:

and to(x)=o, t₁(x)=1, t₂(x)=x²

The first few Tribonacci polynomials are as follows.

t₁(x) = 1

t₂(x) = x²

t₃(x) = x4 +x

t₄(x) = x6 +2x³+1

t₅(x) = x8 +3x⁵+3x²

t₆(x) = x10 +4x⁷+6x4 +2x

t₇(x) = x12 +5x⁹+10x4 +7x³+1

t₈(x) = x14 +6x¹¹+15x8 +16x⁵+6x²…and so on.

Obviously, tn (1) is Just the classical Tribonacci number Tn several basic properties its simple form the Tribonacci polynomials have many interesting properties.

The Generating function of Tribonacci polynomials is defined by

[picture] | (2.1)

Now we will define the Hypergeometric form of generating function of Tribonacci polynomials. In this chapter we shall use following results.

(1)

(2)

(3)

(4)

3. Some Identities of Tribonacci Polynomials

Now we state and prove some Identities of Tribonacci polynomials theorem.

Theorem (3.1).

Proof. we know that the generating function of Tribonacci polynomials is

Theorem (3.2). recurrence relations by the generating function of Tribonacci polynomials we can easily get following recurrence relations.

and

Proof. The generating function of Tribonacci polynomials is

Differentiating (2.1) w.r. to we get

Equating the coefficient of tn+1and tn, we get

and

Theorem (3.3). Prove that and .

Proof.

The Generating function of Tribonacci Polynomials

Put x=0 in eqn (2.1) we get

Equating the coefficients of and on both sides respectively, we get

Theorem (3.4).Prove that

Proof . Differentiating equation (2.1) w.r.t ‘x', we get

Put x=o, we get

on both side respectively, we get

4. Conclusion

In this paper, we introduce some identities Tribonacci polynomials by standard techniques. Some basic identities are obtained by method of generating function. Also some identities are obtained in hyper geometric form.

Acknowledgement

The authors would like to thanks the anonymous referee for carefully reading the paper and for their comments.

References

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[4]   | Catalon, E., Notes surla Theoric des Fractions continuous sur certain series, mem. Acad. R Belgique, 45, (1883) 1-82.
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[11]   | Koshy, T. Fibonacci Lucas numbers with application (willey, New York, (2001).
[12]   | Lucy, Joan slater “Generalized Hypergeometric functions” Cambridge University press (1966).
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[14]   | Srivastava, H. M. and. Manocha, H. L, A Treatise on Generating Functions, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 1984.
[15]   | Swami, M.N.S. problem B-74, The fibonacci Quarterly, 3, (1965), 236. Solution by David Zeitlin, the Fibonacci quarterly, 4(1) (1966), 94.
[16]   | Vorobyou, N.N. The Fibonacci numbers D.C. Health Company, Boston (1963).
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[18]   | Yogesh Kumar Gupta, V. H. Badshah, Mamta Singh, and Kiran Sisodiya, Generalized Additive Coupled Fibonacci Sequences of Third order and Some Identities, International Journal of Innovative Science, Engineering & Technology, Vol. 2 Issue 3, March 2015, 80-85.
[19]   | Yogesh Kumar Gupta, Kiran Sisodiya, Mamta Singh, and Generalization of Fibonacci Sequence and Related Properties, “Research Journal of Computation and Mathematics, Vol. 3, No. 2, (2015), 12-18.
[20]   | Yogesh Kumar Gupta, V. H. Badshah, Mamta Singh, and Kiran Sisodiya, “Diagonal Function of k-Lucas Polynomials.” Turkish Journal of Analysis and Number Theory, vol. 3, no. 2 (2015): 49-52.